Genome rearrangements occur on an evolutionary scale, as well as on a developmental scale in a wide range of differentiating eukaryotic cells. Even in cancer, dramatic DNA rearrangements are frequently observed in somatic cell lineages. This project studies genome rearrangement events using ciliates as model organisms because they undergo massive genome rearrangement during differentiation of an archival germline nucleus into a somatic nucleus that specializes in gene expression. The general mechanism that orchestrates this process of assembly, as recently shown by the PIs, is guided by maternal RNA templates that can be modeled by spatial graphs with rigid 4-valent vertices as a physical representation of the DNA at the time of recombination, and smoothing of the vertices models the actual recombination. The project interlaces experimental and theoretical findings with their mutual reinforcement through 4 specific aims: 1. Understand basic biology of RNA-guided DNA rearrangement and test the role of specific proteins in the rearrangement process. 2. Examine and model unscrambling of complex genome rearrangements, such as overlapping genes. 3. Compare the genome rearrangement maps between different species, and develop mathematical tools for identifying the similarities and differences between genome rearrangements and unscrambling pathways in different species. 4. Develop mathematical techniques to identify and measure structural patterns and complexities of rearrangement processes based on knot and (spatial) graph theory for computing and analyzing gene recombination pathways. The proposed research investigates the basic biology and theory of RNA-based DNA rearrangements on a genome-wide scale, which may offer tremendous insight into the process of gene descrambling and better understanding of factors that influence recombination more generally. Theoretically, spatial graphs are used to model 3D structures of the DNA molecules at the moment of recombination, and methods from knot theory are applied, such as polynomial invariants, linking, and genus ranges.